Hello Madam, I'm sorry to read about his troubles!
The best treatment for the symptoms you describe is nasal hygiene with a saline solution (Hydrasense, Sinus Rinse and others), which can be done as often as needed during the day, and at least 30 minutes before bedtime, followed by suctioning the secretions with a baby fly (depending on the age of the baby). By 'cleaning' the nasal secretions, we can reduce their backward flow, which often causes coughing. It's a bit unpleasant but it works wonders!
For throat irritation, acetaminophen (Tylenol and other brands) is still the best treatment. Encourage him to drink plenty of fluids during the day as well.
I am reassured by the absence of fever, just continue to observe the situation. If he is in good general condition, eating and drinking, even if less than usual, no need to consult.
I hope I've been helpful, don't hesitate if you have any other questions!
Tom Samaha, pharmacist